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Is there a quadratic version of exponential smoothing?

I have a time series - let's say I'm given one new sample (a real value) every second.

I could take a moving average of the last 50
 samples or I could use exponential smoothing of the samples with a time constant of 50
. The resulting graphs of the "output" would be fairly similar. The "current output" of the algorithm assumes that the underlying values are constant plus some noise and the algorithm gives a best-guess of that underlying value.

Or I could do "double exponential smoothing". The "current output" is an estimate of a linear graph. After each sample arrives I have as estimate of the current value and how it is linearly increasing.

What I want is to get a best guess of the quadratic expression for that last 50 samples.

I can find the quadratic expression for that last 50 samples. It's just a regression. But to calculate a regression I'd have to actually look at the last 50 samples - that's too much calculation. I want to do it using some sort of recursive function; rather like exponential smoothing is a cheap way of taking a moving average.

I assumed that "triple exponential smoothing" was what I'm looking for but that's something completely different. It includes a term for "seasonality".

So what is the algorithm? Or what is the Google search term?